272 Prof. Sir W. Thomson on general [Feb. 3, 



first fork, so that their motions shall be equal — that is to say, k(=(J x . 

 Then, putting u to denote the common value of these variables, we have 



_1 d 1 d 1 d 1 du ,. 



U ~V l d^¥ 2 d^'' % 'Y~ 1 d^c¥ i dx {) 



Thus an endless chain or cycle of integrators with disks moved as 

 specified above gives to each fork a motion fulfilling a differential equa- 

 tion, which for the case of the fork of the ^th integrator is equation (4). 

 The differential equations of the displacements of the second fork, third 

 fork, . . .(i — l)th fork may of course be written out by inspection from 

 equation (4). 



This seems to me an exceedingly interesting result ; but though 

 P x , P 2 , P 3 , . . . Pj may be any given functions whatever of cc, the differen- 

 tial equations so solved by the simple cycle of integrators cannot, except 

 for the case of i=2, be regarded as the general linear equation of the 

 order i, because, so far as I know, it has not been proved for any value 

 of i greater than 2 that the general equation, which in its usual form is 

 as follows, 



*S + *£S + - q<J-»=°. < 5 > 



can be reduced to the form (4). The general equation of the form (5), 

 where Q 15 Q„, . . . Q* are any given forms of cc, may be integrated me- 

 chanically by a chain of connected integrators thus : — 



First take an open chain of i simple integrators as described above, 

 and simplify the movement by taking P 1 =P 2 =P 3 = . . . =P 2 =1, so that 

 the speeds of all the disks are equal, and dec denotes an infinitesimal 

 angular motion of each. Then by (2) we have 



dx; d 2 ic i d l ~ 2 Ki „ d l Ki /a . 



?i= ^- 1= ^'---'^ = ^'^ = *? (6) 



"Now establish connexions between the i forks and the ith cylinder, so 

 that 



Qi#i + Q 2 # 2 +..- + Qi-i&-i+Q^i=^ 0) 



Putting in this for g v g 2i &c. their values by (6), we find an equation 

 the same as (5), except that k { appears instead of u. Hence the mechanism, 

 when moved so as to fulfil the condition (7), performs by the motion of 

 its last cylinder an integration of the equation (5). This mechanical 

 solution is complete ; for we may give arbitrarily any initial values to 



**> ffi, 9i-i, --•'&!?,; that is to sa y> t0 



du dhi d'- l u 



dec da? dec 2 



